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three unknowns, for example as in ax2 + 2bxy -f cyz = ?,
where a&c,m are any given integers and it is required to discuss
all integer solutions x, y of the equation. The point to be noted
here is that the problem is stated and is to be solved entirely in
the domain of the rational integers, that is. in the realm of
discrete number. To fit analysis, which is adapted to the investi-
gation of continuous number, to such a discrete problem would
seem to be an impossibility, yet this is what Hermite did.
Starting with a discrete formulation, he applied analysis to the
problem, and in the end came out with results in the discrete
domain from which he had started. As analysis is far more
highly developed than any of the discrete techniques invented
for algebra and arithmetic, Hermite's advance was comparable
to the introduction of modern machinery into a medieval
Hermite had at his disposal much more powerful machinery,
both algebraic and analytic, than any available to Gauss when
he wrote the Disquisitiones Arithmeticae. With Hermite's own
great invention these more modern tools enabled him to attack
problems which would have baffled Gauss in 1800. At one stride
Hermite caught up with general problems of the type which
Gauss and Eisenstem had discussed, and he at least began the
arithmetical study of quadratic forms in any number of un-
knowns. The general nature of the arithmetical theory of
forms' can be seen from the statement of a special problem.
Instead of the Gaussian equation ax* + 2te/ -j- ct/2 = m of
degree ta> in two unknowns (x9 y), it is required to discuss the
integer solutions of similar equations of degree n in 5 unknowns,
where n, s are any integers, and the degree of each term on the
left of the equation is n (not 2 as in Gauss' equation). After
stating how he had seen after much thought that Jacobi's
researches on the periodicity of uniform, functions depend upon
deeper questions in the theory of quadratic forms, Hermite out-
lines his programmes.
'But, having once arrived at this point of view, the problems
- vast enough - which I had thought to propose to myself,
seemed inconsiderable beside the great questions of the general
theory of forms. In this boundless expanse of researches which